Advertisement

Mathematische Annalen

, Volume 370, Issue 1–2, pp 819–839 | Cite as

An equivariant parametric Oka principle for bundles of homogeneous spaces

  • Frank Kutzschebauch
  • Finnur Lárusson
  • Gerald W. Schwarz
Article

Abstract

We prove a parametric Oka principle for equivariant sections of a holomorphic fibre bundle E with a structure group bundle \({{\mathscr {G}}}\) on a reduced Stein space X, such that the fibre of E is a homogeneous space of the fibre of \({{\mathscr {G}}}\), with the complexification \(K^{{\mathbb {C}}}\) of a compact real Lie group K acting on X, \({{\mathscr {G}}}\), and E. Our main result is that the inclusion of the space of \(K^{{\mathbb {C}}} \hbox {-equivariant}\) holomorphic sections of E over X into the space of \(K\hbox {-equivariant}\) continuous sections is a weak homotopy equivalence. The result has a wide scope; we describe several diverse special cases. We use the result to strengthen Heinzner and Kutzschebauch’s classification of equivariant principal bundles, and to strengthen an Oka principle for equivariant isomorphisms proved by us in a previous paper.

Mathematics Subject Classification

Primary 32M05 Secondary 14L24 14L30 32E10 32Q28 

Notes

Acknowledgement

We thank Michael Murray for help with the theory of generalised principal bundles.

References

  1. 1.
    Cartan, H.: Espaces fibrés analytiques. Symposium internacional de topología algebraica, Universidad Nacional Autónoma de México and UNESCO, Mexico City, pp. 97–121 (1958)Google Scholar
  2. 2.
    Forstnerič, F.: Stein manifolds and holomorphic mappings. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 56. Springer, Heidelberg (2011)Google Scholar
  3. 3.
    Forster, O., Ramspott, K.J.: Okasche Paare von Garben nicht-abelscher Gruppen. Invent. Math. 1, 260–286 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Grauert, H.: Holomorphe Funktionen mit Werten in komplexen Lieschen Gruppen. Math. Ann. 133, 450–472 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Heinzner, P.: Geometric invariant theory on Stein spaces. Math. Ann. 289(4), 631–662 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Heinzner, P., Huckleberry, A.: Invariant plurisubharmonic exhaustions and retractions. Manuscr. Math. 83(1), 19–29 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Heinzner, P., Kutzschebauch, F.: An equivariant version of Grauert’s Oka principle. Invent. Math. 119(2), 317–346 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Illman, S.: Existence and uniqueness of equivariant triangulations of smooth proper \(G\)-manifolds with some applications to equivariant Whitehead torsion. J. Reine Angew. Math. 524, 129–183 (2000)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Kutzschebauch, F., Lárusson, F., Schwarz, G.W.: An Oka principle for equivariant isomorphisms. J. Reine Angew. Math. 706, 193–214 (2015)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Lárusson, F.: Model structures and the Oka principle. J. Pure Appl. Algebra 192(1–3), 203–223 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Luna, D.: Slices étales. Sur les groupes algébriques. Soc. Math. France, Paris, pp. 81–105. Bull. Soc. Math. France, Paris, Mémoire 33 (1973)Google Scholar
  12. 12.
    Mandell, M.A., May, J.P.: Equivariant orthogonal spectra and \(S\)-modules. Mem. Am. Math. Soc. 159(755) (2002)Google Scholar
  13. 13.
    Neeman, A.: The topology of quotient varieties. Ann. Math. (2) 122(3), 419–459 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ramspott, K.-J.: Stetige und holomorphe Schnitte in Bündeln mit homogener Faser. Math. Z. 89, 234–246 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Schwarz, G.W.: The topology of algebraic quotients. In: Topological Methods in Algebraic Transformation Groups (New Brunswick, NJ, 1988). Progr. Math., vol. 80, pp. 135–151. Birkhäuser Boston, Boston (1989)Google Scholar
  16. 16.
    Snow, D.M.: Reductive group actions on Stein spaces. Math. Ann. 259(1), 79–97 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Strøm, A.: Note on cofibrations. II. Math. Scand. 22(1968), 130–142 (1969)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  • Frank Kutzschebauch
    • 1
  • Finnur Lárusson
    • 2
  • Gerald W. Schwarz
    • 3
  1. 1.Institute of MathematicsUniversity of BernBernSwitzerland
  2. 2.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia
  3. 3.Department of MathematicsBrandeis UniversityWalthamUSA

Personalised recommendations